I am currently reading the chapter "Fourier Optics" in the book "Fundamentals of Photonics" by Saleh and Teich. However I am not able to follow one specific mathematical derivation.
On page 111 the transfer function of free space is derived $$ H(\nu_x, \nu_y) = \text{exp}(-j 2 \pi d \sqrt{\lambda^{-2} - \nu_x^2 - \nu_y^2}).$$
$d$ is the distance the light travels from the input plane to the output plane. $\lambda$ is the wavelength and $\nu_x$ and $\nu_y$ are the spatial frequency components.
After that this formula is being simplified by using the fresnel approximation, for which it is assumed, that the frequency components $\nu_x$ and $\nu_y$ in the input wave are much smaller than the system bandwidth $\lambda^{-1}$. The resulting approximated transfer function is $$ H_{\text{Fresnel}}(\nu_x, \nu_y) = \text{exp}(j \pi \lambda d (\nu_x^2 + \nu_y^2)) \cdot \text{exp}(-j k d).$$
This still makes sense to me, everything is fine so far. However after that they derive the impulse response of the system by applying the inverse fourier transform to the transfer function $H_{\text{Fresnel}}$. The resulting function is $$h(x,y) \approx \dfrac{j}{\lambda d} \cdot \text{exp}(-j k d) \cdot \text{exp}(-j k \dfrac{x^2+y^2}{2 d}).$$
And honestly, I have absolutely no idea how they come to that expression. The inverse fourier is $$h(x, y) \approx \int_{-\infty}^{+\infty} \int_{-\infty}^{+\infty} H_{\text{Fresnel}}(\nu_x, \nu_y) \cdot \text{exp}(-j 2 \pi (\nu_x x + \nu_y y)) d\nu_x d\nu_y.$$
Small annotation: Out of some reason they flipped the signs in the fourier transform in contrast to the standard notation.
So the core question is: How did they solve this integral? There is a correspondence table at the end of the book, but I have no clue how this should help.
Kind regards